How do I recover data from my presumably dead hard disk

Sounds like you've been bitten by the BSY bug (the drive event log location has been set to an invalid location by an off by one error in the firmware). This contains a reference to your model being affected. At one time, you could send the drive into Seagate and they'd revive it by updating the firmware, you'd end up paying only for shipping. Hopefully they're still doing that. If not, you can do it yourself (another link). If you've got more of theses drives, get the firmware updated while they're still operational.

Update

Here are some additional links. This one seems to be a good general introduction that discusses the cabling / voltage requirements for communicating with the drive. This one has more details / discussion.

If your device is not being recognized by the system, that might indicate a bad controller. That's good, because it means your data is likely still safe on the disks themselves. But it's also bad, because you'll need to use a data recovery firm that can replace the controller in a clean room environment. But at least there is a relatively high chance of their success.

This is worth trying before paying for recovery or opening up the drive. Something that has worked for me is to place the drive in a a Ziplock type bag and remove as much air as you can. Place the drive in a freezer for several hours. If possible, connect the drive to a known good system with a USB adapter or external case( put case in bag too) and see if you can read. This has worked a number of times for me.

Sometimes a longer stay in freezer has helped.

Others have found this works Freeze Drive A quick Google found may videos and other articles. IBM server training used to mention this as a fix to try.

My Seagate external drive started failing two weeks ago. It got very warm, so I unplugged it and let it cool, and it would work fine again for a couple of days. Eventually I began to hear clicking and knew mechanical failure was near. I managed to archive what I needed most from it before it completely quit and refused to be recognized.

I'd heard about the freezer trick for years but never believed it. At this point I figured, what have I got to lose, so I put plastic kitchen wrap around the drive and let it sit in the freezer overnight. To my surprise, after a couple of minutes, the drive's folders began to show up one by one. I immediately began copying files to a fresh drive, which worked great for about 15 minutes, until the HD warmed up and quit again.

Seeing that temperature is a factor, and having heard that you can do this trick four or five times before it totally fails, I brought a portable refrigerator/freezer into my office so that the drive could sit in it and stay cold on my next attempt. Unfortunately, all I've gotten so far is an "install new software" prompt, but it doesn't see its software to install itself. Had I been able to keep it cool on my first attempt, I might have been able to recover everything, but I'll never know.

So the freezer trick worked for me, but not long at all. For the first time ever I'm considering a cloud backup service.